How do you find lim (sqrt(x+2)-sqrtx)/(sqrt(x+3)-sqrtx) as x->oo?

1 Answer
Cesareo R.
Feb 11, 2017

2/3

Explanation:

(sqrt(x+2)-sqrt(x))/(sqrt(x+3)-sqrt(x)) = (sqrt(x+2)-sqrt(x))/(sqrt(x+3)-sqrt(x)) ((sqrt(x+3)+sqrt(x))/(sqrt(x+3)+sqrt(x)))
=((sqrt(x+2)-sqrt(x))(sqrt(x+3)+sqrt(x)) )/3
Calling now f(x)=((sqrt(x+2)-sqrt(x))(sqrt(x+3)+sqrt(x)) )/3 we have

f(x)((sqrt(x+2)+sqrt(x))/(sqrt(x+2)+sqrt(x))) =
2/3((sqrt(x+3)+sqrt(x)) /(sqrt(x+2)+sqrt(x)))=(2/3)(sqrt(1+3/x)+1)/(sqrt(1+2/x)+1)

Finally

lim_(x->oo)(sqrt(x+2)-sqrt(x))/(sqrt(x+3)-sqrt(x))=lim_(x->oo)(2/3)(sqrt(1+3/x)+1)/(sqrt(1+2/x)+1) = 2/3

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